The package has been updated to work with version 8.7 of Sage.

A DeprecationWarning: is_RingHomomorphism() has been fixed.
(see http://trac.sagemath.org/23204 for details) and the code
has been updated at severak other places (see diff for details).

Thanks to Frédéric Chapoton for his help on this commit!

src/sage/schemes/chow/all.py

@@ -1,11 +1,11 @@
 # code exports
+from __future__ import absolute_import
 
-from ring import ChowRing, is_chowRing, PointChowRing
-from scheme import ChowScheme, is_chowScheme, PointChowScheme
-from finite_ring_extension import FiniteRingExtension
-from sheaf import SHom, SEnd, Sheaf, is_sheaf
-from bundle import Bundle, TrivialBundle, is_bundle
-from blowup import Blowup
-from morphism import is_chowSchemeMorphism
-from library.all import *
-
+from .ring import ChowRing, is_chowRing, PointChowRing
+from .scheme import ChowScheme, is_chowScheme, PointChowScheme
+from .finite_ring_extension import FiniteRingExtension
+from .sheaf import SHom, SEnd, Sheaf, is_sheaf
+from .bundle import Bundle, TrivialBundle, is_bundle
+from .blowup import Blowup
+from .morphism import is_chowSchemeMorphism
+from .library.all import *

src/sage/schemes/chow/blowup.py

@@ -79,6 +79,7 @@
     sage: P5.<k> = Proj(5, 'k')
     sage: f = P2.hom([2*h], P5)
     sage: g = Blowup(f)
+    sage: TestSuite(g).run(skip=["_test_category","_test_pickling"])
 
 AUTHORS:
 
@@ -98,8 +99,7 @@
 
 from sage.all import QQ
 from sage.rings.polynomial.term_order import TermOrder
-# noinspection PyUnresolvedReferences
-from sage.matrix.all import matrix
+from sage.matrix.constructor import matrix
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.schemes.chow.finite_ring_extension import FiniteRingExtension
 from sage.schemes.chow.library.proj import ProjBundle
@@ -136,7 +136,7 @@ class Blowup(ChowSchemeMorphism):
     def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
                  domain_name=None, codomain_name=None,
                  latex_domain_name=None, latex_codomain_name=None):
-        """
+        r"""
         Construct the class `:class:Blowup`.
 
         INPUT:
@@ -203,7 +203,7 @@ def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
 
         # Get the finite ring extension associated to Hf: HY --> HX
         if verbose:
-            print "Computing finite ring extension for Hf: HY --> HX..."
+            print("Computing finite ring extension for Hf: HY --> HX...")
 
         A_f = FiniteRingExtension(Hf, var_name=var_name)
 
@@ -212,7 +212,7 @@ def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
         #######################################################################
 
         if verbose:
-            print "Computing Exceptional locus..."
+            print("Computing Exceptional locus...")
         # Get the normal bundle N_{X/Y} from the exact sequence:
         #       0 --> TX --> f^*(TY) --> N_{X/Y} --> 0
         TX, TY = X.tangent_bundle(), Y.tangent_bundle()
@@ -232,7 +232,7 @@ def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
         # viewed as HY-module under Hf.
 
         if verbose:
-            print "Computing Blowup..."
+            print("Computing Blowup...")
         vnms = A_f.nvs() + Y.variable_names()    # Strings
         mds1 = tuple(d + 1 for d in A_f.mds())   # Add 1 to the module degrees
         degs = mds1 + Y.degs()
@@ -253,13 +253,13 @@ def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
         rels = rels + R.ideal(new_rels)
 
         # 3) Add module relations of HX as HY module (eg presentation matrix)
-        mod_matrix = T * A_f.prm().apply_map(lambda xx: R(xx))
+        mod_matrix = T * A_f.prm().apply_map(lambda t: R(t))
         new_rels = mod_matrix.list()
         rels = rels + R.ideal(new_rels)
 
         # 4) Multiplicative relations from m_i * m_j
         mm = [m * n for m in A_f.mgs() for n in A_f.mgs()]
-        pd = A_f.push_down(mm).apply_map(lambda xx: R(xx))
+        pd = A_f.push_down(mm).apply_map(lambda t: R(t))
         mu = matrix(s, s, (T * pd).list())
         new_rels = (T.transpose() * T - last_nv * mu).list()
         rels = rels + R.ideal(new_rels)
@@ -269,7 +269,7 @@ def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
         z = HXX.cover_ring()(proj_var_name)
         u = (z ** codim).reduce(HXX.defining_ideal())
         uc = [HX(str(u.coefficient({z: d}))) for d in range(codim)]
-        pd = A_f.push_down(uc).apply_map(lambda xx: R(xx))
+        pd = A_f.push_down(uc).apply_map(lambda t: R(t))
         M = matrix(codim, 1, [(- last_nv) ** d for d in range(codim)])
         new_rels = (mu * pd * M - T.transpose() * (- last_nv) ** codim).list()
         rels = rels + R.ideal(new_rels)
@@ -286,9 +286,9 @@ def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
         top = S.dual().chern_classes()[codim - 1].lift()
         top = top.reduce(HXX.defining_ideal())
         topc = [HX(str(top.coefficient({z: d}))) for d in range(codim)]
-        pd = A_f.push_down(topc).apply_map(lambda xx: R(xx))
+        pd = A_f.push_down(topc).apply_map(lambda t: R(t))
         JX = matrix(s, 1, f.lowerstar(list(A_f.mgs()), verbose=verbose))
-        JX = JX.apply_map(lambda xx: R(xx))
+        JX = JX.apply_map(lambda t: R(t))
         new_rels = (mu * pd * M - JX).list()
         rels = rels + R.ideal(new_rels)
 
@@ -302,7 +302,7 @@ def __init__(self, f, var_name='e', proj_var_name='z', verbose=False,
                         name=codomain_name, latex_name=latex_codomain_name)
 
         if verbose:
-            print "Computing tangent bundle of blowup..."
+            print("Computing tangent bundle of blowup...")
 
         SY = YY.hom(Y.gens(), Y)
         YY = YY.base_change(SY)

src/sage/schemes/chow/bundle.py

@@ -190,4 +190,5 @@ def TrivialBundle(X, r):
     """
     return Bundle(X, r, [1] + [0] * min(r, X.dimension()))
 
+
 PointChowScheme.sheaves["tangent"] = Bundle(PointChowScheme, 0, [1])

src/sage/schemes/chow/finite_ring_extension.py

@@ -77,20 +77,17 @@
 # ****************************************************************************
 
 from sage.all import QQ
-
+from sage.matrix.constructor import matrix
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.rings.quotient_ring import QuotientRing_generic
 from sage.rings.polynomial.term_order import TermOrder
-# noinspection PyUnresolvedReferences
-from sage.rings.polynomial.multi_polynomial_ring_generic import is_MPolynomialRing
-from sage.schemes.chow.ring import Kernel2
-# noinspection PyUnresolvedReferences
 from sage.libs.singular.function import singular_function, lib as singular_lib
-# noinspection PyUnresolvedReferences
-from sage.matrix.all import matrix
+from sage.rings.polynomial.multi_polynomial_ring_generic import \
+    is_MPolynomialRing
+
+from sage.schemes.chow.ring import Kernel2
 
 
-# noinspection PyCallingNonCallable
 class FiniteRingExtension(QuotientRing_generic):
 
     def __init__(self, f, var_name='z'):
@@ -181,9 +178,9 @@ def __init__(self, f, var_name='z'):
         ff = f  # A priori A, B are MPolynomial.
         # Check the ring A and get the rings AA and AI:
         if is_MPolynomialRing(A):
-            AA, AI = A, A.ideal(0)
+            AA = A
         elif isinstance(A, QuotientRing_generic):
-            AA, AI = A.cover_ring(), A.defining_ideal()
+            AA = A.cover_ring()
             ff = f.morphism_from_cover()
         else:
             err = "Domain is not (quotient of) Multivariate Polynomial Ring."
@@ -335,61 +332,9 @@ def __init__(self, f, var_name='z'):
         Q = matrix(R, IX.ngens(), 1, L[1])
         M = matrix(R, 1, s)
         M[0, s - 1] = 1
-        self._prm = (P + (Q * M)).transpose().apply_map(lambda xx: A(str(xx)), A)
+        self._prm = (P + (Q * M)).transpose().apply_map(lambda x: A(str(x)), A)
         QuotientRing_generic.__init__(self, CC, CI, c_vars)
 
-    def __iter__(self):
-        r"""
-        Return an iterator through the elements of ``self``.
-        Not implemented in general.
-
-        EXAMPLES::
-
-            sage: A.<y> = ChowRing('y', 1, 'y^6')  # P5
-            sage: B.<x> = ChowRing('x', 1, 'x^3')  # P2
-            sage: f = A.hom([2*x], B)
-            sage: F = FiniteRingExtension(f)
-            sage: sage.schemes.chow.finite_ring_extension.FiniteRingExtension.__iter__(F)
-            Traceback (most recent call last):
-            ...
-            NotImplementedError: FiniteRingExtension does not support iteration
-        """
-        raise NotImplementedError("FiniteRingExtension does not support iteration")
-
-    def characteristic(self):
-        r"""
-        Return the characteristic of this ring, which is the same
-        as the characteristic of its base ring.
-
-        EXAMPLES::
-
-            sage: A.<y> = ChowRing('y', 1, 'y^6')  # P5
-            sage: B.<x> = ChowRing('x', 1, 'x^3')  # P2
-            sage: f = A.hom([2*x], B)
-            sage: F = FiniteRingExtension(f)
-            sage: F.characteristic()
-            0
-        """
-        return self.base_ring().characteristic()
-
-    def krull_dimension(self):
-        """
-        Return the Krull dimension of this commutative ring.
-
-        EXAMPLES::
-
-            sage: A.<y> = ChowRing('y', 1, 'y^6')  # P5
-            sage: B.<x> = ChowRing('x', 1, 'x^3')  # P2
-            sage: f = A.hom([2*x], B)
-            sage: F = FiniteRingExtension(f)
-            sage: F.krull_dimension()
-            Traceback (most recent call last):
-            ...
-            NotImplementedError
-
-        """
-        raise NotImplementedError
-
     def ann(self):
         r"""
         Return the annihilator `Ann_A(B)` of `B` seen as an `A`-module via `f`.
@@ -611,7 +556,7 @@ def push_down(self, v):
         M = matrix(self, 1, s)
         M[0, s - 1] = 1
 
-        return ((P + (Q * M)).transpose()).apply_map(lambda xx: self(str(xx)))
+        return ((P + (Q * M)).transpose()).apply_map(lambda x: self(str(x)))
 
     def psi(self):
         r"""

src/sage/schemes/chow/library/grass.py

@@ -22,7 +22,6 @@
 from sage.schemes.chow.scheme import PointChowScheme
 from sage.schemes.chow.sheaf import SHom, Sheaf
 from sage.schemes.chow.scheme import ChowScheme
-# noinspection PyUnresolvedReferences
 from sage.rings.integer import is_Integer
 
 
@@ -87,7 +86,7 @@ def GrassBundle(A, B, chern_class='c', names=None, name=None, latex_name=None):
     #
 
     EZ = Bundle(Z, n, [str(c) for c in E.chern_classes()])  # p^{*}E
-    QZ = Bundle(Z, r, [str(1)] + new_vars)  # if s==k : c_{s+1} = ... = c_{r} = 0
+    QZ = Bundle(Z, r, [str(1)] + new_vars)  # if s==k : c_{s+1} =...= c_{r} = 0
     SZ = Bundle(Z, k, (EZ - QZ).chern_classes()[0: k + 1])
     QZ = Bundle(Z, r, (EZ - SZ).chern_classes()[0: r + 1]) if s == k else QZ
 

src/sage/schemes/chow/library/proj.py

@@ -30,8 +30,7 @@ def Proj(n, hyperplane_class='h', names=None, name=None, latex_name=None):
     - ``n`` -- An integer, the dimension of the projective space.
     - ``hyperplane_class`` - An (optional) name for the hyperplane class
     - ``name`` -- An optional string, the name of the ChowScheme
-    - ``latex_name``-- An optional string, the latex representation of the
-                       ChowScheme
+    - ``latex_name``-- An optional string, the latex representation of the ChowScheme
 
     OUTPUT:
 
@@ -52,17 +51,17 @@ def Proj(n, hyperplane_class='h', names=None, name=None, latex_name=None):
                  name=name, latex_name=latex_name)
 
 
-def ProjBundle(E, hyperplane_class='h', names=None, name=None, latex_name=None):
-    """
+def ProjBundle(E, hyperplane_class='h',
+               names=None, name=None, latex_name=None):
+    r"""
     Return the *Proj* of a bundle E.
 
     INPUT:
 
     - ``E`` -- A sheaf on a ChowScheme
     - ``hyperplane_class`` - An (optional) name for the hyperplane class
     - ``name`` -- An optional string, the name of the ProjBundle
-    - ``latex_name``-- An optional string, the latex representation of the
-                       ProjBundle
+    - ``latex_name``-- An optional string, the latex representation of the ProjBundle
 
     OUTPUT:
 

src/sage/schemes/chow/library/twisted_cubics.py

@@ -83,14 +83,16 @@
 
 The incidence variety can be obtained in a two-step process
 `I_W={\mathbb P}(K_1^*)\xrightarrow{v} {\mathbb P}(W)\xrightarrow{u} S`, where
-`0\to K_1\to u^*W\to L_1\to 0` is the tautological sequence on `{\mathbb P}(W)`.
+`0\to K_1\to u^*W\to L_1\to 0` is the tautological sequence on
+`{\mathbb P}(W)`.
 
 This is implemented in :meth:`incidence_variety`.
 
 Also, let `0\to K_2\to v^*K_1^*\to L_2\to 0` be the tautological sequence on
 `I_W`. Since the Chow ring of `H^*(X_W)` is generated by the Chern classes of
 `E` and `F`, the embedding `f:I_W\to X_W` is determined as soon as one can
-identify `f^*E` and `f^*F`. According to Ellingsrud and Strømme , these bundles are
+identify `f^*E` and `f^*F`. According to Ellingsrud and Strømme,
+these bundles are
 `f^*E= K_1\otimes  L_2^*` and `f^*F=K_2\otimes  L_2^*\otimes  \det(K_1)`.
 
 Finally the morphism `f` is implemented in
@@ -198,20 +200,19 @@ def variety_of_nets_of_quadrics(W, name=None, latex_name=None):
 
     A_f = FiniteRingExtension(Af)
     ideal_list = A_f.push_down(J).list()
-    I = AY.ideal([str(l) for l in ideal_list])
+    IRel = AY.ideal([str(l) for l in ideal_list])
 
-    # I contains the relations. Cleaning up:
+    # IRel contains the relations. Cleaning up:
 
     # a) Simplify::
 
-    # noinspection PyUnresolvedReferences
     from sage.libs.singular.function import singular_function
     simplify = singular_function('simplify')
 
     to = TermOrder('wdegrevlex', tuple([1, 2, 3, 1, 2] + S_degs))
     C = PolynomialRing(QQ, 5 + S.ngens(),
                        names=['e1', 'e2', 'e3', 'f1', 'f2'] + S_vars, order=to)
-    CI = C.ideal([str(x) for x in I.gens()])
+    CI = C.ideal([str(x) for x in IRel.gens()])
     CI = simplify(CI, 15, ring=C)
 
     # b) e1 = f1::
@@ -307,17 +308,17 @@ def incidence_variety(W, name=None, latex_name=None):
     K1 = PW.sheaves["universal_sub"]
     L1 = PW.sheaves["universal_quotient"]
 
-    I = ProjBundle(K1.dual(), 'x', name=name, latex_name=latex_name)
-    v = I.base_morphism()
-    I.sheaves["K1"] = v.upperstar(K1)
-    I.sheaves["L1"] = v.upperstar(L1)
+    IV = ProjBundle(K1.dual(), 'x', name=name, latex_name=latex_name)
+    v = IV.base_morphism()
+    IV.sheaves["K1"] = v.upperstar(K1)
+    IV.sheaves["L1"] = v.upperstar(L1)
 
     w = u * v
-    I = I.base_change(w)
-    I.sheaves["K2"] = I.sheaves["universal_sub"]
-    I.sheaves["L2"] = I.sheaves["universal_quotient"]
+    IV = IV.base_change(w)
+    IV.sheaves["K2"] = IV.sheaves["universal_sub"]
+    IV.sheaves["L2"] = IV.sheaves["universal_quotient"]
 
-    return I
+    return IV
 
 
 def map_incidence_to_nets_of_quadrics(W, domain_name=None, codomain_name=None,
@@ -334,14 +335,19 @@ def map_incidence_to_nets_of_quadrics(W, domain_name=None, codomain_name=None,
         sage: f = map_incidence_to_nets_of_quadrics(W)
     """
     S = W.chowscheme()
-    I = incidence_variety(W, name=domain_name, latex_name=latex_domain_name)
-    IE = I.sheaves["K1"] * I.sheaves["L2"].dual()
-    IF = I.sheaves["K2"] * I.sheaves["L2"].dual() * I.sheaves["K1"].determinant()
+
+    IV = incidence_variety(W, name=domain_name, latex_name=latex_domain_name)
+    IV_K1 = IV.sheaves["K1"]
+    IV_K2 = IV.sheaves["K2"]
+    IV_L2 = IV.sheaves["L2"]
+
+    IE = IV_K1 * IV_L2.dual()
+    IF = IV_K2 * IV_L2.dual() * IV_K1.determinant()
     X = variety_of_nets_of_quadrics(W, name=codomain_name,
                                     latex_name=latex_codomain_name)
     ie1 = IE.chern_classes()[1]
     ie2 = IE.chern_classes()[2]
     ie3 = IE.chern_classes()[3]
     if2 = IF.chern_classes()[2]
-    f = I.hom([ie1, ie2, ie3, if2] + list(S.gens()), X)
+    f = IV.hom([ie1, ie2, ie3, if2] + list(S.gens()), X)
     return f